Higher Order Bifurcations of Limit Cycles

This paper shows that asymmetrically perturbed, symmetric Hamiltonian systems of the formx=y,y=±(x±x3)+λ1y+λ2x2+λ3xy+λ4x2y,with analyticλj( )=O( ), have at most two limit cycles that bifurcate for small ≠0 from any period annulus of the unperturbed system. This fact agrees with previous results of Petrov, Dangelmayr and Guckenheimer, and Chicone and Iliev, but shows that the result of three limit cycles for the asymmetrically perturbed, exterior Duffing oscillator, recently obtained by Jebrane and o adek, is incorrect. The proofs follow by deriving an explicit formula for thekth-order Melnikov function,Mk(h), and using a Picard–Fuchs analysis to show that, in each case,Mk(h) has at most two zeros. Moreover, the method developed in this paper for determining the higher-order Melnikov functions also applies to more general perturbations of these systems